Abstract
In this article, the authors consider the Schrödinger type operator \(L:=-\mathrm{div}(A\nabla )+V\) on \({\mathbb {R}}^n\) with \(n\ge 3\), where the matrix A satisfies uniformly elliptic condition and the nonnegative potential V belongs to the reverse Hölder class \(RH_q({\mathbb {R}}^n)\) with \(q\in (n/2,\,\infty )\). Let \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\,\infty )\) be a variable exponent function satisfying the globally \(\log \)-Hölder continuous condition. When \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (1,\,\infty )\), the authors prove that the operators \(VL^{-1}\), \(V^{1/2}\nabla L^{-1}\) and \(\nabla ^2L^{-1}\) are bounded on variable Lebesgue space \(L^{p(\cdot )}({\mathbb {R}}^n)\). When \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\,1]\), the authors introduce the variable Hardy space \(H_L^{p(\cdot )}({\mathbb {R}}^n)\), associated to L, and show that \(VL^{-1}\), \(V^{1/2}\nabla L^{-1}\) and \(\nabla ^2L^{-1}\) are bounded from \(H_L^{p(\cdot )}({\mathbb {R}}^n)\) to \(L^{p(\cdot )}({\mathbb {R}}^n)\).
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Acknowledgements
Junqiang Zhang is very grateful to his advisor Professor Dachun Yang for his guidance and encouragements. Junqiang Zhang is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2018QS01) and the National Natural Science Foundation of China (Grant No. 11801555). Zongguang Liu is supported by the National Natural Science Foundation of China (Grant No. 11671397).
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Communicated by Mieczysław Mastyło.
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Zhang, J., Liu, Z. Some estimates of Schrödinger type operators on variable Lebesgue and Hardy spaces. Banach J. Math. Anal. 14, 336–360 (2020). https://doi.org/10.1007/s43037-019-00020-6
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DOI: https://doi.org/10.1007/s43037-019-00020-6